Numerical Analysis

Zhiping Li

LMAM and School of Mathematical SciencesPeking University

Lecture 20: Monte Carlo Methods

~E|§Metropolis

~E|— Cþ!©ÄÚéóCþ

Cþ

Cþg´: |^5£Xþ¤®,?nÅCþ¿©/C0ÅCþ5/0?nÅCþOØ. ~X, ∫ 1

0f (x)dx =

∫ 1

0[f (x)− g(x)]dx +

∫ 1

0g(x)dx .

d Monte Carlo , Xi , i = 1, · · · ,N, i .i .d . ∼ U [0, 1], µ

IN(f ) =1

N

N∑i=0

[f (Xi )− g(Xi )] + I (g),

ùp I (g) ®. e Var(f − g) < Var(f ), KþªÒÑ«.

2 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

~E|— Cþ!©ÄÚéóCþ

Cþ~

P r(x) = eσx , σ > 0 ~ê, ÄÈ©

I (f ) =

∫ +∞

−∞

1√2π

(1 + r(x))−1e−x2

2 dx .

5¿

(1 + r(x))−1 ≈ h(x) ,

1, x ≤ 0;

0, x > 0,

òÈ©U¤

I (f ) =1√2π

∫ +∞

−∞

((1 + r(x))−1 − h(x)

)e−

x2

2 dx +1

2,

2|^IO©Ù­5ÄO1Ü©È©. ùph(x) åCþ^.

3 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

~E|— Cþ!©ÄÚéóCþ

©Ä

©Ä(¡AÛ©) g´òÈ©«¿©¤eZf«, ¿3zf«þ©OA^ Monte Carlo . TÐ?3u: zf«þ¼ê±z, ÏdN´¦^CþÚ­5Ä. =B^ü¿©ÚÄ Monte Carlo , . ~X, ÄÈ©

I (f ) =

∫ 1

0f (x)dx .

ò« Ω = [0, 1] © M °µΩk = [k−1M , k

M ], k = 1, · · · ,M;

i.i.d. X(k)i ∼ U(Ωk), i = 1, · · · , n, k = 1, · · · ,M, N = nM

ÅCþ; -

InM(f ) =M∑k=1

1

nM

n∑i=1

f (X(k)i ) =

1

N

M∑k=1

n∑i=1

f (X(k)i ).

5¿§dCz3uÄ´Uf«©O?1.

4 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

~E|— Cþ!©ÄÚéóCþ

ü©Ä InM(f ) þ

P fk = M∫

Ωkf (x)dx = |Ωk |−1

∫Ωk

f (x)dx = Ef (X (k)), Kk

EInM(f ) =1

nM

M∑k=1

n∑i=1

fk =M∑k=1

1

Mfk =

M∑k=1

∫Ωk

f (x)dx = I (f ),

Var(InM(f )) = E |InM(f )− I (f )|2

=1

N2

M∑k,l=1

n∑i,j=1

E[(f (X

(k)i )− fk

)(f (X

(l)j )− fl

)]=

1

N2

M∑k=1

[nM

∫Ωk

(f (x)− fk

)2dx]

=1

N

∫Ω

|f (x)− f (x)|2dx , 1

Nσ2s ,

Ù¥ f (x) = fk , ∀x ∈ Ωk , k = 1, · · · ,M.

5 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

~E|— Cþ!©ÄÚéóCþ

ü©Ä InM(f ) o¬~

·Kµ éü©Ä InM(f ), k

σs ≤ σ ,∫

Ω|f (x)− I (f )|2dx

−1/2,

Ò= fk = I (f ), ∀1 ≤ k ≤ M, ¤á.

y²µ g¼ê gk(c) ,∫

Ωk|f (x)− c |2dx , k = 1, · · · ,M, ©O

3 g ′k(c) = 2∫

Ωk(f (x)− c)dx = 0, = c = fk ?

. Ïd, gk(fk) ≤ gk(I (f )), Ò= I (f ) = fk ¤á.·Ky.

6 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

~E|— Cþ!©ÄÚéóCþ

éóCþ (antithetic variables method)

éóCþ´«AÏé½Âäk½é¡5, ȼêäkAÏ5/¤OAÏE|.

ȼê f (x) ½Â [0, 1], Kk±e(ص

·Kµ XJ f (x) ´üN, X ∼ U [0, 1], K'Xê÷v

Cov(f (X ), f (1− X )) =

∫ 1

0(f (x)− I (f ))(f (1− x)− I (f ))dx ≤ 0.

y²µ Cov(f (X ), f (1− X )) =

∫ 1

0

f (x)f (1− x)dx −[ ∫ 1

0

f (x)dx]2

=

∫ 1

0

∫ 1

0

f (x)f (1− x)dxdy −∫ 1

0

∫ 1

0

f (x)f (y)dxdy

=

∫ 1

0

∫ 1

0

f (x)f (1− x)dxdy −∫ 1

0

∫ 1

0

f (x)f (1− y)dxdy

7 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

~E|— Cþ!©ÄÚéóCþ

éóCþ£Y¤

=

∫ 1

0

dx

∫ x

0

f (x)(f (1−x)−f (1−y))dy+

∫ 1

0

dx

∫ 1

x

f (x)(f (1−x)−f (1−y))dy

=

∫ 1

0

dx

∫ x

0

f (x)(f (1−x)−f (1−y))dy+

∫ 1

0

dy

∫ y

0

f (x)(f (1−x)−f (1−y))dx

=

∫ 1

0

dx

∫ x

0

(f (x)− f (y))(f (1− x)− f (1− y))dy ≤ 0.

تd f üNÚ (x − y)((1− x)− (1− y)) = −(x − y)2 ≤ 0.

- IN , 12N

∑Ni=1[f (Xi ) + f (1− Xi )], k EIN = I (f ). d±þ·K

Var(IN) = E |IN−I (f )|2 =1

4N2E( N∑

i=1

(f (Xi )−I (f ))+(f (1−Xi )−I (f )))2

=1

2N

[Var(f ) + Cov(f (X )− I (f ), f (1− X )− I (f ))

]≤ 1

2NVar(f ).

8 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

~E|— Cþ!©ÄÚéóCþ

êÈ© Monte Carlo Ø&Ý

êÈ© Monte Carlo Ø eN = |IN(f )− I (f )| E,´ÅCþ, ·kþØE |eN |2 = 1

NVar(f (X )) 9Ø

Ï"EeN ≤√E |eN |2 =

√Var(f (X ))

N . ddO eN

Var(eN) = E (eN − EeN)2 = E (|eN |2)− (EeN)2 ≤ 1

NVar(f (X )).

u´dMarkov تP(|X | ≥ ε) ≤ ε−αE |X |α, ∀α > 0

P(|eN | ≥ ε) ≤ ε−2E |eN |2 =1

ε2NVar(f (X )),

P(|eN−EeN | ≥ ε) ≤ ε−2E (eN−EeN)2 =1

ε2Var(eN) ≤ 1

ε2NVar(f (X )).

5551: ε = C

√Var(f (X ))

N, C > 1, KkP(|eN | ≥ ε) ≤ C−2. AO, P(∩m

i=1(|e iN | ≥ ε)) ≤ C−2m .

5552: Markov ت9Ùy²ë p.168, 5VÇØ6§ÛÖ?ͧ®ÆÑ, 2006c.

9 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

Metropolis — «ê¼ó Monte Carlo

·3OÅþ¢y Monte Carlo Ä?Ö´±pÇ)Ñl½©ÙÅê. k©Ù, cÙ´êépmþ©Ù, %éJ^·c¡ùü3OÅþ)Ñlù©Ù£¤Åê.

Metropolis <Ñ«|^ê¼ó=£VÇÝ)Ñl,A½©ÙÅê. Metropolis , ¡ê¼óMonte Carlo , 3ÚOÔnXn²þ.È©£½¦Ú¤¥k2A^.

·± Ising .~, 0 Metropolis . Ún Ising .´^>fg^ïÄc^áC5..

10 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

c^áC5 Ising .

M ¬:iüm, z¬:þk>f. ^ σi = ±1L«1 i :þ>fg^ ↑ ½ ↓. ¤k M >fg^ σ = (σ1, · · · , σM) ÑT.XÚ*.

k M ¬: Ising .k 2M *. XÚ?u* σ , XÚSU±^Uþ¼ê H(σ) L«

H(σ) = −J∑〈i ,j〉

σiσj , σk ∈ 1,−1, k = 1, · · · ,M,

Ù¥ 〈i , j〉 L«¦Ú=u;:é, = |i − j | = 1 . c^á J > 0, c^á J < 0.

11 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

Ising .*VÇ©ÙXn²þ

Ising .* σ 3XÚ¥ÑyVÇÑl Gibbs ©Ù,ÙVÇÝL« 1

ZMexp−βH(σ), Ù¥ β = (kBT )−1, kB

Boltzmann ~ê, T ýé§Ý, ZM =∑

σ exp−βH(σ) ©¼ê.

u´, XÚ÷*ÚOþ, ~X/üâf²þSU0±ÏLXÚ*/²þ0£Xn²þ¤¦

UM =1

M

∑σ

exp−βH(σ)ZM

H(σ) ,1

M〈H(σ)〉.

Monte Carlo 8I: )¤VÇÝ 1ZM

exp−βH(σ) i.i.d. ÅCþS σ(i)Ni=1, O 〈H(σ)〉 ≈ 1

N

∑Ni=1 H(σ(i)).

12 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

Metropolis Äg

3vk|^¹e, XÚl?Щ*GÑu, ²LãmüC, o¬ªu²ï. ù,z*E¬p=, XÚoN?uIJï¥, XÚ÷*ÔnþØmCz. Ïd, Xn²þ±^m²þO. =

〈H(σ)〉 ≈ 1

N

N∑i=1

H(σ(i)),

Ù¥GS σ(i)Ni=1 ´lЩGÑuÏL·/Ôn5K0£~X©f-E5K¤)àê¼ó. ÚOÔn¥òXÚù«m²þum²þ5¡XÚH5.

Metropolis Ò´ÄuùÄg, ÏLE·àê¼ó±VÇÝ 1

ZMexp−βH(σ) ²­©Ù=£V

ÇÝ5)mGS.

13 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

Ising .*VÇmþ=£VÇÝ

- Ω = σ| M ¬þ¤k*, F Ω f8¤)¤ σ ê, é S ∈ F ½Â¯ S u)VÇ

P(S) =∑σ∈S

exp−βH(σ)ZM

,

K (Ω,F ,P) ¤VÇm. Ω ¥¤kØÓ*σ êNt = 2M , òØÓ*σ ÑyVÇ 1

ZMexp−βH(σ)

U*,«üSª σ(i)Nti=1 ü¤Nt 1þ, Pπ.

½ÂVÇm (Ω,F ,P) þ=£VÇÝ P = (pij)Nt×Nt , Ù¥pij l* σ(i) σ(j) =£VÇ, ÷v: (1) pij ≥ 0, ∀i , j= 1, · · · ,Nt ; (2)

∑Ntj=1 pij = 1, ∀i = 1, · · · ,Nt .

8Iµ À=£VÇÝP, ¦ª)¤ÅCþSÑlGibbs ©Ù.

14 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

*VÇmþ=£VÇÝkÚAk5

1 dÅÝ5£1¤!£2¤, ÚÝØ¥ Gerschgorin½n(G.H.Golub & C.F. van Loan, ”Matrix Computation”) 1 =£VÇÝ P A, AmAþ (1, · · · , 1)T .

½Âµ XJ A ∈ RNt×Nt ÷veãöµ

Nt = 1 A = 0;

Nt ≥ 2, 3Ý Q ∈ RNt×Nt 9ê 1 ≤ r < Nt ,¦

QTAQ =

(B C0 D

)Ù¥ B ∈ Rr×r , 0 ´"Ý,

K¡ A Ý, ÄK, ¡ÙØÝ.

15 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

*VÇmþ=£VÇÝkÚAk5£Y¤

½Âµ XJ3g,ê τ , ¦ê¼ó=£VÇÝ P ÷vPτ > 0, = Pτ zÑ´ê, K¡Tê¼ó, ¡=£VÇÝ P .

5µ ݽ´Ø, ØÝؽ´. ~X, P, Ù¥ p11 = p22 = 0, p12 = p21 = 1.

2 5µ =£VÇÝP AT´Ý. d± P =£VÇÝê¼ó´§=l?GÑu3kÚ=£So¬±VÇ?ÛÙ§G.

16 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

*VÇmþ=£VÇÝkÚAk5£Y¤

Perron-Frobenius½nµXJ A ∈ RNt×Nt ´KØÝ, K

A Ì» ρ(A) > 0 ´ A ü­A;

Au ρ(A) Aþ¤k©þþ"ÓÒ;

Ø3AuÙ§AKAþ (©þþK).

5µ dPerron-Frobenius ½nÚGerschgorin ½n 1 =£VÇÝP ü­A, 3Au 1 !mAþ, Ø3Ù§KAþ. ® (1, · · · , 1)T ´=£VÇÝP Au 1 mAþ§ P Au 1 AþK´¤¢=£VÇÝP ØC©Ù.

(Perron-Frobenius ½ny²:Mä, ”ÝOnØ”)

17 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

*VÇmþ=£VÇÝkÚAk5£Y¤

½Âµ éê¼ó=£VÇÝP, XJ©Ù µ = µP, K¡ µ ´Tê¼óØC©Ù£¡´=£VÇÝP ØC©Ù¤.

3 * σ(i) ÑyVÇ 1ZM

exp−βH(σ(i)) ü¤Nt 1

þ π AT´=£VÇÝP ØC©Ù.

XÚIJï, üØÓ*σ Úσ′ ÑyVÇÒ©O´π(σ) = 1

ZMexp−βH(σ) Úπ(σ′) = 1

ZMexp−βH(σ′),

π ´ P ØC©Ù¿X π(σ′) =∑

σ π(σ)P(σ → σ′).

½Âµ XJê¼ó÷v

π(σ)P(σ → σ′) = π(σ′)P(σ′ → σ)

K¡Tê¼ó÷v[²ï^, ½¡Tê¼ó´_.

18 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

Metropolis =£VÇÝäk5

[²ï^¿Xê¼ó²­©Ù*mp='Xé¡5£5¿ù¿=£VÇÝé¡5¤. 3ÚOÔn¥ù«é¡523.

nþ¤ã, Metropolis ¦=£VÇÝP äkXe5µ

1 =£VÇÝP ÷v: (i) pij ≥ 0, ∀i , j = 1, · · · ,Nt ; (ii)∑Ntj=1 pij = 1, ∀i = 1, · · · ,Nt .

2 =£VÇÝP ´Ý. d±P =£VÇÝê¼ó´£¤.

3 * σ(i) ÑyVÇ 1ZM

exp−βH(σ(i)) ü¤Nt 1

þ π ´=£VÇÝP ØC©Ù.

4 d=£VÇÝP ½Âê¼ó÷v[²ï^.

19 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

äk5 (1)-(4)ê¼óÂñ5

dê¼ónØ, äk5 (1)-(4)ê¼óäkXeÂñ5µ

½nµ ê¼óäk5 (1)-(4), ¼ê g(σ) ÷v E |g(σ)| =∑σ∈Ω π(σ)|g(σ)| <∞. KédTê¼ó)¤l?ЩG

σ(0) ÑuGS σ(1), σ(2), · · · , σ(n), · · · Ñk1

n

n∑i=1

g(σ(n))→∑σ∈Ω

π(σ)g(σ), n→∞, a.s.

ùp a.s. (almost surely) Âñ´VÇ 1 Âñ.

5µ AO/§äk5 (1)-(4)ê¼ókXeÂñ5µ

1

n

n∑i=1

H(σ(i))→ 〈H(σ)〉, n→∞, a.s.

20 / 22

Lecture 20: Monte Carlo Methods

~E|§Metropolis

Metropolis

Metropolis Ø%— ¢y5 (1)-(4)

=£VÇÝP k N2t ©þ, 5 (1), (3) Ú(4) ©OJÑ

Nt , Nt Ú Nt(Nt − 1)/2 å^. , kõgdÝ5ÀJ=£VÇÝP, ØÓÀJéAØÓ.

Metropolis ÏLê[A½÷v[²ï^ê¼óuÐL§, l )GS σ(1), σ(2), · · · , σ(n). uÐL§zg¢y©*dÕáüÚµ1Ú, )ýÀ σ′; 1Ú, ä´ÄÉ£±õVÇɤT#G,XJÉ, K σ(n+1) = σ′, ÄK, σ(n+1) = σ(n).

ØÓýÀG)ÑØÓ, ùOE,ÝU¬ØÓ.

21 / 22

SKÔµ4; þÅSKÔµ4.

Thank You!